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Mirrors > Home > MPE Home > Th. List > moabs | Structured version Visualization version GIF version |
Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2650. (Revised by BJ, 14-Oct-2022.) |
Ref | Expression |
---|---|
moabs | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑)) | |
2 | nexmo 2619 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | |
3 | id 22 | . . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑) | |
4 | 2, 3 | ja 188 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑) |
5 | 1, 4 | impbii 211 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1776 ∃*wmo 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 |
This theorem depends on definitions: df-bi 209 df-ex 1777 df-mo 2618 |
This theorem is referenced by: mo3 2644 mo4 2646 moeu 2664 dffun7 6381 wl-mo3t 34811 |
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